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I am wondering how to get the steady-state for the following Euler equation. I know that we can get rid of time in subscripts. However, here I have a constant (a) to the power of $t$. Does anyone know if there is a way to get rid of $t$ in power? Or can I just consider $a^t$ a new constant, say $\bar{a}$?

$$ \frac{1}{c_{t}}=a^t \beta E_t \Big[(1+r_{t+1})\dfrac{1}{c_{t+1}}\Big] $$

MJD
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Giordano
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  • Is the utility funktion $U_t(c_t)=\log(c_t)$? – callculus42 Feb 19 '21 at 14:00
  • yes, the utility function is $log(c_t)$ – Giordano Feb 19 '21 at 14:06
  • Do you know if $r_{t+1}$ or $c_{t+1}$ are random variables? If not, then you can omit the expected value operator for the first step. – callculus42 Feb 19 '21 at 14:26
  • I know that I can rewrite the steady-sated as $1=a^t \beta (1+r)$. I just dropped $c_t$ and $c_{t+1}$ since they are identical in steady-state. Now my question is what should do with $a^t$? $a$ is a constant but to the power of t. I suppose I cannot drop $t$ from $a^t$ and just take it as $a$. What is the trick in this case? – Giordano Feb 19 '21 at 14:54
  • $a^{t^}$ is an option. Here $t^$ is the time where the function is maximized. – callculus42 Feb 19 '21 at 15:18

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